Our estimate for the fusion timescales is based on the adiabatic condition, required for the system to remain in its ground state during the full protocol. This condition should be fulfilled throughout the fusion and reference protocols, except close to point 1 in Fig. REF  (a), where the ground state is close to doubly degenerate and the operation should be non-adiabatic with respect to the small remnant splitting. The adiabatic condition is given by [1]}
\(\mbox{max}_k\left[f_k(t)\right]\ll 1,\) 
